[PPT] - Proton decay matrix elements on lattice Jun-Sik Yoo 1 1 Department of PowerPoint Presentation

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Introduction Effective operator Matrix elements on lattice Future projects Conclusion Reference

Proton decay matrix elements on lattice

Jun-Sik Yoo 1

1Department of Physics and Astronomy

Stony Brook University

2019 Lattice x Intensity Frontier Workshop, BNL, Sept. 23-25, 2019

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Introduction Effective operator Matrix elements on lattice Future projects Conclusion Reference

Introduction

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Proton Decay

Process p − → Π + ℓ Baryon Number Violation Haven’t been observed. Lifetime > 1034 years Does proton even decay?

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Experimental bound

Current proton decay bound in SK, (ABE et al., 2018)

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Proton Decay

Process p − → Π + ℓ Baryon Number Violation Haven’t been observed. Lifetime > 1034 years Motivated by Baryon Asymmetry Possible explanation by GUT, SUSY-GUT

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Effective operators

Four-fermion effective operators

p − → Π + ℓ Hadronic states : Nonperturbative computation required.

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Effective operator

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GUT, SUSY-GUT

GUT Symmetry group to be G ⊃ SU(3)C ⊗ SU(2)L ⊗ U(1)Y ♣ Coupling unification ♣ Baryon asymmetry SUSY-GUT ♣ Superpartners to particles ♣ Better unification at higher scale

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GUT,SUSY-GUT

(a) d=4 operator (b) d=5 operator (c) d=6 operator

Possible BV operators in (SUSY-)GUT

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GUT,SUSY-GUT

(d) ∼ ΛGUT (e) ∼ ΛSUSY (f) ∼ ΛEW

Proton decay operator at different scales

Model parameters come into Wilson coefficients (a) Yqq, Yql, Yud, Yue (b) MHC (c) m˜

l, m˜ q, triangle loop integrals, ...

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Effective operators

Figure 1: Four-fermion effective operators

Effective operator : OΓΓ′ = (qq)Γ(qℓ)Γ′, (XY )Γ = (X TCPΓY ) C := (Charge Conjugation Matrix) Π¯ ℓ|pGUT ∼ C ΓΓΠ¯ ℓ|OΓΓ′|pSM = C ΓΓ ¯ vℓΠ|(qq)ΓPΓ′q|p, where C ΓΓ′ is a wilson coefficient, Π is a meson, and p is a proton.

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Decay rate

The decay rate Γ is calculated from the hadronic matrix element, Π(p′)|OΓΓ′(q)|N(p, s) = ¯ vℓPΓ′

W ΓΓ′

(q2) − i/ q mN W ΓΓ′

1

(q2)

uN(p, s)

= ¯ vℓPΓ′W ΓΓ′ (q2)uN(p, s) + O(ml/mN) ¯ vℓuN(p, s) (1) where Π a meson, N a nucleon, and W0,1 decay form factor(AOKI et al., 2000). Then the decay rate is Γ

p → Π + ¯

ℓ

= (m2

p − m2 Π)2

32πm3

p

I

CIW I

p → Π + ¯

ℓ

2

. (2)

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Matrix elements on lattice

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Introduction Effective operator Matrix elements on lattice Future projects Conclusion Reference

Hadronic states

2 4 6 8 10 12 t/a 0.575 0.600 0.625 0.650 0.675 0.700 0.725 0.750 log(C(t)/C(t+1)) [ 1 13] [ 2 13] [ 3 13] [ 4 13] log (C2pt) vs. t

pK = [0, 1, 1]pmin Correlation function C 2pt

K (t,

p) =

x

ei

p· x0|JK(t,

x)J†

K(0,

0)|0 = asymptotic states + ... Excited states at early times Ground state at late times

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Three-point function

(Meson)-(Decay Operator)-(Proton)

C 3pt(t, t′) =

x,

x′

ei(

p′· x′− q· x)0|JΠ(x′)O(x) ¯

JN(x0)|0 = Π( p′)|O|N( p) × C 2pt

Π (t′ − t,

p′) √ZΠ Tr[PC 2pt

p

(t, p)]

Zp

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Lattice settings

Lattice paramters

lattice size 243 × 64 × 24 gauge action Iwasaki-DSDR fermion DWF β 1.633 lattice cutoff a−1 = 1GeV ml a 0.00107 mha 0.0850 mπa 0.1387 mK a 0.5051 mres 0.00228 mπL 3.3 Deflated CG 2000+1000 AMA 32+1 Ncfg 102 lattice size 323 × 64 × 32 gauge action Iwasaki-DSDR fermion DWF β 1.75 lattice cutoff a−1 = 1.37GeV ml a 0.0001 mha 0.0450 mπa 0.1046 mK a 0.3602 mres mπL 3.3 Deflated CG 2000+250 AMA 32+1 Ncfg 20(and counting) JS Yoo Proton Decay 2019 Lattice x Intensity Frontier Workshop, / 41

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Kinematic Choice

Energy-momentum conservation & q2 ∼ 0

0.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20 q2a2 0.0 0.5 1.0 1.5 2.0 2.5 3.0

|p| pmin

012 002 111 011 001

π

0.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20 q2a2 0.0 0.5 1.0 1.5 2.0 2.5 3.0

|p| pmin

001 011 111

K

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Matrix elements

1 2 3 4 5 6

t/a

0.045 0.050 0.055 0.060 0.065 0.070 0.075

W0a2

2 = 2.355

p = [0, 1, 1] < K0|(us)LuL|p >

decay form factor W LL

0 (p → K 0e+) at tsep = 8

pp = [0, 0, 0]pmin, pK = [0, 1, 1]pmin plateau fit t=3–5 AMA 32+1, 102 configs

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Matrix elements

Decay matrix elements w/ different src-sink separation {8,9,10}

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Matrix elements

Decay matrix elements w/ different src-sink separation {8,9,10}

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Comparison with earlier work

Bare value, but multiplicative renormalization only − → ratio can be compared with renormalized values W norm =

W ΓΓ′

(Channel) W ΓΓ′ (K +|(ds)ΓuΓ′|p)

(3)

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Comparison with earlier work

0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 K0|(us)LuL|p K0|(us)LuR|p K + |(us)LdL|p K + |(us)LdR|p K + |(ud)LsL|p K + |(ud)LsR|p K + |(ds)LuL|p K + |(ds)LuR|p

+ |(ud)LdL|p + |(ud)LdR|p

This Study Aoki:2017

Comparison with earlier study, (AOKI et al., 2017)

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Comparison with earlier work

0.0 0.5 1.0 1.5 2.0 K0|(us)LuL|p K0|(us)LuR|p K + |(us)LdL|p K + |(us)LdR|p K + |(ud)LsL|p K + |(ud)LsR|p K + |(ds)LuL|p K + |(ds)LuR|p

+ |(ud)LdL|p + |(ud)LdR|p

This Study Aoki:2017

Comparison with earlier study, (AOKI et al., 2017)

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Comparison to earlier work

Stat. [%]

(This study) Stat.[%] (Aoki:2017) Chiral extrapol.[%] a2 [%] ∆Z [%] K 0|(us)LuL|p 2.80 3.5 3.1 5.0 8.1 K 0|(us)LuR|p 1.77 2.8 2.8 5.0 8.1 K +|(us)LdL|p 3.32 4.4 7.5 5.0 8.1 −K +|(us)LdR|p 2.24 3.7 3.5 5.0 8.1 K +|(ud)LsL|p 2.13 3.0 3.9 5.0 8.1 −K +|(ud)LsR|p 2.12 3.2 1.6 5.0 8.1 −K +|(ds)LuL|p 2.01 2.8 2.1 5.0 8.1 −K +|(ds)LuR|p 2.96 3.6 2.7 5.0 8.1 −π+|(ud)LdR|p 6.17 3.4 2.7 5.0 8.1 π+|(ud)LdR|p 4.62 3.0 2.7 5.0 8.1

Left : Comparison of statistical errors. Right: Systematic errors in chiral extrapolation, O(a2), ∆Z ( (AOKI et al., 2017))

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Renormalization

0.5 1.0 1.5 2.0 2.5 3.0 3.5 2(GeV2) 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

diagonal smom

LL LL RL RL A(LV) A(LV)

Three quark op. renormalization 32+1 AMA, 20 configs, Landau gauge RI-SMOM scheme Two-loop Matching (GRACEY, 2012) fit region (pa)2 = 1 − 3.5GeV2 UMS←latt

LL

(µ = 2GeV) = 0.67(2) UMS←latt

RL

(µ = 2GeV) = 0.68(1)

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Future projects

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Future projects

Proton decay matrix elements can be investigated futher to see: ♣ Induced Nucleon Decay from Dark matter ♣ Vector meson channels from proton decay ♣ Three body decay channel from proton decay

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Vector meson channels

♣ Same computation with different Γ structures ♣ Different form factor decomposition ♣ Asymptotic vector meson channel should be there. K ∗i(Q)ℓ(p′)|Od=6|p(p, s) = ǫi

µ¯

vc

ℓ [F1γ5γµ + F2iγ5σµνQν + F3γ5Qµ

+ F ′

1γµ + F ′ 2iσµνQν + F ′ 3Qµ]uN

(4)

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Vector meson channels

Lattice paramters(BOYLE et al., 2016)

lattice size 243 × 64 × 16 gauge action Iwasaki fermion DWF β 2.13 lattice cutoff a−1 = 1.78GeV ml a 0.0005 mha 0.04 mπ 339.6MeV mπL 4.568 JS Yoo Proton Decay 2019 Lattice x Intensity Frontier Workshop, / 41

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Vector meson channels

4 6 8 10 12 t/a 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38

Eff. Mass (GeV)

4 5 6 7 8 9 10 11 t/a 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

Eff. Mass (GeV)

4 5 6 7 8 9 10 11 t/a 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Eff. Mass (GeV)

4 6 8 10 12 t/a 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Eff. Mass (GeV)

Two point fcn. mπ = 345(6)MeV, p-val = 0.08 mρ = 916(46)MeV, p-val = 0.00 mσ = 595(40)MeV, p-val = 0.99 mp = 1.315(16)GeV, p-val=0.924

*No disconnected diagram is included.

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Three body decay

A few reasons to compute three body decay : ♣ Resonant to vector meson channels : p → K ∗ + ¯ ℓ → (Kπ)¯ ℓ ♣ Decay rate ratio (Γ(p → ππe+)/Γ(p → πe+)) estimates to ∼ 24–150% (WISE; BLANKENBECLER; ABBOTT, 1981) ♣ Prime channel of next generation experiment ♣ Numerically cheapest among three body decay channels

DUNE proton decay efficiency

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Future projects

Induced Nucleon Decay model (DAVOUDIASL et al., 2010) ♣ DM can annihilate the nucleon ♣ Leff ∼ (1/Λ3)uRdRdRYRΦ+ h.c. ♣ Π(p′)|OΓΓ′(q)|N(p, s) = PΓ′

W ΓΓ′

(q2) + mY

mN W ΓΓ′ 1

(q2)

uN(p, s)

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Conclusion

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Conclusion

♣ Proton decay matrix elements on the two lattice ensemble with chiral fermions at physical scale ♣ Three quark op. non perturbative renormalization ♣ New channels(vector meson, three body) ♣ Induced nucleon decay by DM

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Reference

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Reference

Reference ABE, K. et al. Hyper-Kamiokande Design Report. 2018. AOKI, S. et al. Nucleon decay matrix elements from lattice

QCD. Phys. Rev., D62, p. 014506, 2000.

AOKI, Y. et al. Improved lattice computation of proton decay matrix elements. Phys. Rev., D96, n. 1, p. 014506, 2017. BOYLE, P. A. et al. Low energy constants of SU(2) partially quenched chiral perturbation theory from Nf =2+1 domain wall

QCD. Phys. Rev., D93, n. 5, p. 054502, 2016.

DAVOUDIASL, H. et al. Hylogenesis: A Unified Origin for Baryonic Visible Matter and Antibaryonic Dark Matter. Phys.

Rev. Lett., v. 105, p. 211304, 2010.

GRACEY, J. A. Three loop renormalization of 3-quark

perators in QCD. JHEP, v. 09, p. 052, 2012.

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Introduction

Figure 2: Energy scale of search, Zoltan Ligeti

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Baryon asymmetry Nonzero net baryon number

nB−¯ nB nγ

∼ 10−10 Sakharov’s conditions ♣ At least one B violating process ♣ C- and CP-violation ♣ interactions outside of thermal equilibrium

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Matrix elements

Define the ratio R3(t, t′) =

C 3pt(t,t′) C 2pt

Π (t′−t,

p′)Tr[PC 2pt

p

(t, p)]

√ZΠ

Zp.

As t → ∞, R3(t, t′) → Π(p′)|OΓΓ′(q)|N(p, s), giving decay form factors W0,1(q2) Tr[R3PLP4] = W ΓL

0 (q2) − iq4

mN W ΓL

1 (q2).

Tr[R3PLiP4γj] = qj mN W ΓL

1 (q2)

Momentum transfer is chosen to be q2 ∼ 0 : p = q + p′

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Other computations

Figure 3

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